Parametric and non-parametric distributions


We believe that there is a very useful distinction to be made between model-based parametric and empirical non-parametric distributions. By 'model-based', I mean a distribution whose shape is borne of the mathematics describing a theoretical problem. For example: an exponential distribution is the direct result of assuming that the rate of decay of x is proportional to x; a Lognormal distribution is derived from assuming that ln[x] is Normally distributed, etc.

By 'empirical distribution' we mean a distribution whose mathematics is defined by the shape that is required. For example, a Triangle distribution is defined by its minimum, mode and maximum values; a Histogram distribution is defined by its range, the number of classes and the height of each class. The defining parameters for general distributions are features of the graph shape. Empirical distributions include: Cumulative, Discrete, Histogram, Relative, Triangle and Uniform.

Those distributions that fall under the 'empirical distribution' or non-parametric class are intuitively easy to understand, extremely flexible and are therefore very useful. Model-based or parametric distributions require a greater knowledge of the underlying assumptions if they are to be used properly. Without that knowledge, the analyst may find it very difficult to justify the use of the chosen distribution type and to gain peer confidence in his model. He will probably also find it difficult to make alterations should more information become available.

We are keen advocate of using non-parametric distributions. I believe that parametric distributions should only be selected if either: a) the theory underpinning the distribution applies to the particular problem, b) it is generally accepted that a particular distribution has proven to be very accurate for modelling a specific variable without actually having any theory to support the observation, or c) the distribution approximately fits the expert opinion being modelled and the required level of accuracy is not very high. The last scenario would include use of the PERT, Exponential, Lognormal, Normal, and Pareto distributions.

 

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